Sharp Bounds for Sums Assocated to Graphs of Matrces at the WaldenFest Roland Specher Unverstät des Saarlandes Saarbrücken, Germany
Academc famly tree of Wlhelm von Waldenfels Schürmann Köng Wese Köng Specher Waldenfels Skede
Academc famly tree of Wlhelm von Waldenfels Schürmann Köng Wese Köng Specher and sons Waldenfels Skede
Hedelberg... at the good old tmes... last century
Some thngs we learned... thnkng s hard work...
Some thngs we learned... thnkng s hard work...
Some thngs we learned... look for a (hdden) conceptual structure behnd your complcated concrete problem...
Queston What s the asyptotc behavour n N of 1,..., 2m some constrants on equalty of ndces 1 2 t (2) 3 4 t (m) 2m 1 2m wth gven matrces T k = ( t (k) ) N,=1
Examples t,=1 T =1 t T,,k=1 t(2) k t(3) k T 1 T 2 k,,k,l=1 t(2) k t(3) l T 2 T 1 k l
That s easy? Okay, so look on ths: 1,..., 8 =1 1 2 t (2) 3 2 t (3) 3 4 t (4) 4 4 t (5) 5 3 t (6) 2 5 t (7) 6 5 t (8) 6 5 t (9) 6 6 t (10) 7 5 t (11) 8 7 t (12) 8 7 4 T 4 3 1 T 2 T 2 1 T 5 T 6 T 7 T 6 11 T 10 T 8 8 T 9 T 12 7 5
Problem For a gven drected graph G (multple edges and loops allowed), wth matrces attached to the edges, denote the correspondng sum by S G (N) Queston: What s the optmal bound r(g) n S G (N) N r(g) m k=1 T k
Optmal asymptotcs n N? 4 T 4 3 1 T 2 T 2 1 T 5 T 6 T 7 T 6 11 T 10 T 8 8 T 9 T 12 7 5 1 1 t (2) 2 3 t (3) 2 3 t (4) 4 4 t (5) 4 5 t (6) 3 2 t (7) 5 6 t (8) 5 6 t (9) 5 6 t (10) 6 7 t (11) 5 8 t (12) 7 8 N??? 7 12 =1 T
Motvaton such sums appear and have to be asymptotcally controlled n calculatons of moments of products of random (Wgner) matrces and determnstc matrces Yn + Krshnaah, 1983 Ba (+ Slversten), 1999 (2006) Mngo + Specher, 2012 JFA (asymptotc freeness of Wgner and determnstc matrces) Male, 2012 traffcs
Example =1 t T Then or =1 t =1 T = N T =1 t = Tr(T ) = N tr(t ) N T
Example =1 t T Then or =1 t =1 T = N T =1 t = Tr(T ) = N tr(t ) N T
Example,,k=1 t(2) k t(3) k T 1 T 2 k,,k=1 t(2) k t(3) k = Tr(T 1T 2 ) = N tr(t 1 T 2 ) N T 1 T 2 N T 1 T 2
Example t,=1 T trval estmate:,=1 t,=1 T = N 2 T But we can do better...
t,=1 T... wth usng the vectors we can wrte e := (0,..., 1,..., 0), e := (1, 1,..., 1) and thus,=1 t =,=1 e, T e = e, T e snce,=1 t e, T e e 2 T = N T e = N
t,=1 T... wth usng the vectors we can wrte e := (0,..., 1,..., 0), e := (1, 1,..., 1) and thus,=1 t =,=1 e, T e = e, T e snce,=1 t e, T e e 2 T = N T e = N
,,k,l=1 Example t(2) k t(3) l T 2 T 1 k l t(2) k t(3) l =,,k,l=1,,k,l e, T 1 e e, T 2 e k e, e l }{{} e e,t 2 e k e l = e, T 1 e e e }{{} V :C N C N C N T 2, e e = e, T 1 V (T 2 ) e e
,,k,l=1 Example t(2) k t(3) l T 2 T 1 k l,,k,l=1 t(2) k t(3) l =,,k,l e, T 1 e e, T 2 e k e, e l }{{} e e,t 2 e k e l = e, T 1 e e e }{{} V :C N C N C N T 2, e e = e, T 1 V (T 2 ) e e
,,k,l=1 Example t(2) k t(3) l T 2 T 1 k l,,k,l=1 t(2) k t(3) l =,,k,l e, T 1 e e, T 2 e k e, e l }{{} e e,t 2 e k e l = e, T 1 e e e }{{} V :C N C N C N T 2, e e = e, T 1 V (T 2 ) e e
,,k,l=1 Example t(2) k t(3) l T 2 T 1 k l,,k,l=1 t(2) k t(3) l =,,k,l e, T 1 e e, T 2 e k e, e l }{{} e e,t 2 e k e l = e, T 1 e e e }{{} V :C N C N C N T 2, e e = e, T 1 V (T 2 ) e e
,,k,l=1 t(2) k t(3) l T 2 T 1 k l,,k,l=1 t(2) k t(3) l = e, T 1 V (T 2 ) e e e e e }{{} e e T 1 V T }{{} 2 =1 = N 3/2 T 1 T 2 snce V : C N C N C N, e e s an sometry, and thus V = 1. e, f = 0, f
General structure In ths example, our sum S(N) s gven as nner product where S(N) ˆ= output, stuff nput each nput and each output vertex contrbutes factor N 1/2 nternal vertces do not contrbute, summaton over them corresponds to matrx multplcaton or, more general, partal sometres
Note that one can also modfy,,k=1 t(2) k t(3) k T 1 T 2 k to an nput-output form: T 1 T 2 t k
General strategy Modfy gven graph to equvalent nput-output graph then each nput and output vertex gves factor N 1/2 Man problems: Can we always do such a modfcaton? If yes, how can we recognze how many nput/output vertces we need?
Asymptotcs determned by structure of F(G) F(G) = forrest of two-edge-connected components 4 T 4 4 3 1 T 2 T 2 1 T 5 2 = 3 = 5 = 6 5 T 6 T 7 T 6 11 T 10 T 8 8 T 9 T 12 7 1 7 = 8
Theorem [Mngo, Specher, JFA 2012] We have the followng optmal estmate: :V [N] e E t (e) t(e), s(e) N 1 2 #leaves of F(G) e E T e (Trval leaves,.e., only vertces of trval trees, count twce!)
Optmal asymptotcs n N! 4 T 4 4 3 1 T 2 T 2 1 T 5 2 = 3 = 5 = 6 5 T 6 T 7 T 6 11 T 10 T 8 8 T 7 12 T9 1 7 = 8 1 1 t (2) 2 3 t (3) 2 3 t (4) 4 4 t (5) 4 5 t (6) 3 2 t (7) 5 6 t (8) 5 6 t (9) 5 6 t (10) 6 7 t (11) 5 8 t (12) 7 8 N 3/2 7 12 =1 T
Replace by equvalent nput-output problem 4 T 4 4 T 4 4 3 3 1 T 2 T 2 1 T 5 2 T t 2 T 5 5 T 6 T 7 6 T 6 T 7 T 6 11 T 10 T 8 8 T 9 T 12 7 5 T 1 T 10 T8 t 5 7 6 T 9 T 11 T 12 1 8
Addng addtonal vertces 4 T 4 T 4 4 3 T t 2 T 5 5 T 5 2 T 6 T 7 6 T t 2 T 7 I T N 6 6 T 9 T 9 T 1 T 10 T8 t 5 7 T 11 T 12 T 10 T t 8 T 1 T 11 T 12 1 8
H n V n,2 : H n H H T 4 T 4 V 2,1 : H H H T 5 V 1,2 : H H H T 5 T t 2 T 7 I T N 6 V 1,1 V 1,3 : H H) H (H H H) T t 2 T 6 T 7 V 2,1 V 1,1 V 1,2 : (H H) H H H H (H H) T 9 T 9 V 1,1 V 1,1 V 2,1 : H H (H H) H H H T 1 T8 t T 10 T 11 T 12 T t 8 V 1,1 V 2,1 : H (H H) H H T 10 V 1,1 V 1,2 : H H H (H H) T 1 T 11 T 12 H out H out V 1,out V 2,out : H (H H) H out H out
Theorem [Mngo, Specher, JFA 2012] We have the followng optmal estmate: :V [N] e E t (e) t(e), s(e) N 1 2 #leaves of F(G) e E T e (Trval leaves,.e., only vertces of trval trees, count twce!)
Happy Brthday, Wlhelm!!!